 Research article
 Open Access
Modeling elasticplastic buckling of sandwich axisymmetric shells: on the limits of "shell" models and analytical solutions
 Alain G Combescure^{1}Email author
https://doi.org/10.1186/2213746712
© Combescure; licensee Springer. 2014
Received: 1 March 2013
Accepted: 3 May 2013
Published: 29 January 2014
Abstract
Background
The objective of this paper is to answer the question: "Can a 'shell' model always be used to predict the elastic buckling of a shell?"
Method
This paper shows that such a model leads to significantly overestimated critical loads in the case of sandwich shells and gives an explanation for this overestimation.
Results
A dependable model is proposed and applied to a few structures of revolution, for which it is shown that shell analyses are sometimes overly on the unsafe side.
Conclusion
Of course, in such cases, 3D analysis is possible, but the associated computation cost is several orders of magnitude higher than that of the Fourier series analysis proposed in this paper.
Keywords
 Critical Load
 Critical Pressure
 Fourier Mode
 Core Layer
 Tangent Modulus
Background
Computational modeling techniques to analyze the buckling of anisotropic and multilayered shells are very well developed, so that further advances may not seem necessary. However the quest for lighter structures, which has induced many studies for decades, has led to the development of sandwich structures in order to manufacture robust, yet lightweight, mechanical parts. Some of these parts are subjected to compression and their buckling strength must be assessed. The prediction of the critical buckling loads of these structures seems to be simple: it suffices to model these objects as multilayered shells, thus leading to an immediate prediction of their critical loads. However, a comparison of the critical loads predicted from calculations with experimental results reveals significant differences. Some remarkable works on buckling [1–3] were published at the turn of the century, and quasianalytical solutions have been proposed for cylinders in axial compression [4]. These solutions take into account two types of buckling: global shell buckling and local skin buckling (wrinkling). Many works dealing with the study of crashworthiness optimization (e.g. [5–7]) are also available. Some authors have also worked on the interaction between the two types of buckling [8–11] and shown that in some cases this interaction leads to a critical load which is smaller than that predicted by calculating the critical loads of each mode independently. The objective of the present work is to examine the finite element modeling of this problem in the case of axisymmetric structures and propose a modeling strategy which leads to good results in all situations. The particular case of plastic buckling is also analyzed in depth.
The paper is structured as follows:

first, the modeling of elasticplastic buckling and its application to structures of revolution is reviewed;

then, three sample applications are presented(a cone under internal pressure, a cone under external pressure, and a sphere under external pressure).
Method : A short review of the modeling of the elasticplastic buckling of structures of revolution
Elastic buckling of structures of revolution
where u,v,w denote respectively the axial, circumferential and normal displacements of the shell. r is the distance to the axis of revolution, R the axial curvature of the shell, and ϕ the angle with direction z (+z being the axis of revolution).
Finally, buckling analysis consists of two steps: first the calculation of the axisymmetric prestress and then a series of buckling calculations for each Fourier mode j. The critical load and the associated critical mode correspond to the Fourier mode j which leads to the smallest load.
The shell finite elements chosen for the calculations are plane shell elements ($\frac{1}{R}=0$) which possess only four degrees of freedom per node (three translations plus the rotation about the circumferential direction) [15].
The volume elements chosen are 8node isoparametric axisymmetric elements with three Fourier degrees of freedom.
Nonlinear buckling of structures of revolution
Nonlinear instability may develop in any non linear analysis: for a given load level, they are characterized by the existence of more than one equilibrium state. For an axisymmetric structure the unstable state can be either axisymmetric or nonaxisymmetric. If the structure remains axisymmetric, wellknown incremental calculation techniques with control (such as the arclength control method [17, 18]) can generally be used to predict the critical load, except when the loading is exactly orthogonal to the instability mode. However, if the instability mode is not axisymmetric, this instability cannot be predicted using incremental analysis alone. In that case, there are two possible modeling strategies for predicting instability.

The first strategy consists in meshing the structure of revolution in 3D and using this mesh to seek the instability. However, one should note that if the mesh satisfies the symmetry of revolution and the loading is axisymmetric the 3D analysis leads to an axisymmetric deformed shape and, thus, "misses" the nonaxisymmetric instability.
In this case, one must either introduce a defect (by perturbing the mesh of the structure or the loading) or use a mesh which does not satisfy the symmetry of revolution, hoping that this "imperfection" will solicit the instability sufficiently for it to develop "naturally."
One can also use the specific technique of axisymmetric elements with nonaxisymmetric defects ([16, 19]).

The second strategy consists in calculating the nonlinear response of the structure of revolution (which, thus, remains axisymmetric), then checking the stability of each resulting nonlinear state k. In such a stability study, for each load step k, one examines all the possible Fourier modes: the instability point is the first step k whose load multiplier is equal to 1. This Fourier analysis with uncoupled modes is still possible even though the response of the structure is nonlinear (in terms of geometry or material) because the preloads remain axisymmetric.
The geometrically nonlinear case
In the case of a pure geometrically nonlinear response, one calculates the axisymmetric equilibrium states using standard nonlinear incremental techniques. Thus, one obtains a sequence of m equilibrium states, denoted C_{ k }, which are characterized by two variable fields: the displacement field between the initial structure and the current structure u_{ k }, and the equilibrated Cauchy stress field σ_{ k }. Once these m states have been found, one studies their stability. In order to do that, one calculates the stability of the deformed shape which corresponds to each state C_{ k }by examining all the possible Fourier modes to find the mode j which leads to the smallest critical load ${\lambda}_{j}^{k}$. The first state k which satisfies ${\lambda}_{j}^{k}=1$ is the critical state: then, buckling occurs following Fourier mode j. One should stress the fact that this buckling is usually associated with a loss of symmetry of revolution (if j ≠ 0).
The elasticplastic case
Results and discussions
Now let us apply the buckling models to the prediction of elastic and elasticplastic buckling for three types of sandwich shells:

a halfsphere fixed at the base and subjected to uniform external pressure;

three cone frustra fixed at their bases and subjected to uniform external pressure;

the same three cone frustra, this time subjected to uniform internal pressure.
Geometries and materials
All the shells considered consisted of three layers. The two skins were the same thickness (t = 1 mm) and the core layer was filled with a material which was generally weaker than the skin. Three core layers with thicknesses e equal to 1, 9 and 49 mm, respectively (making their overall thicknesses equal to 3, 11 and 51 mm) are examined. The chosen generic geometries and the properties of the materials, which were assumed to be isotropic, are defined below:
The halfsphere
The halfsphere had a radius R equal to 1 m and was fixed at the base.
The cones
Three types of cones, with angles 30°, 45° and 60° from the horizontal, were studied. All three had the same radius at the base R_{ min }equal to 1 m and the same maximum radius R_{ max }equal to 2 m. The bases were fixed. Thus, 9 cases under internal pressure and 9 cases under external pressure were studied.
The materials
The materials were assumed to be elastic or perfectly elasticplastic. The yield strain was 0.001. The same Poisson’s coefficient is chosen for all materials (equal to 0.3). The Young’s modulus of the skins was E_{ skin } = 200, 000 MPa. For the core layer of the sandwich shells, decreasing values of the Young’s modulus were chosen among the following list: E_{ core } = 200,000 MPa, 20,000 MPa, 2,000 MPa, 200 MPa and 20 MPa. For the nonlinear analyses, the moduli E_{ core } = 200,000 MPa, 2,000 MPa and 20 MPa alone were studied. Hereafter, the ratio $\frac{{E}_{\mathit{\text{core}}}}{{E}_{\mathit{\text{skin}}}}$ will be denoted β.
The finite element models
All these cases were modeled as axisymmetric problems in two ways. The first model will be referred to as the COQMULT model. The midline was meshed using p + 1 nodes, i.e. p elements. Each layer was assumed to have its own offset, its own thickness and its own material. The second model will be referred to as the QUA8 model. The thickness was represented by 3 quadrangular isoparametric elements with 8 nodes and 9 integration points. The generatrix was meshed using q elements. Thus, the final mesh contained 3q elements and 10q + 7 nodes. The stability analysis was carried out in the Fourier basis by seeking the Fourier mode leading to the lowest critical load.
In the case of the QUA8 mesh of the cone, rectangular elements were chosen. Thus, the mesh represented the shell’s geometry exactly. A preliminary convergence study was performed for each model in order to predict the linear buckling pressure of a single layer under external or internal pressure. This study enabled us to choose the mesh size for each case. The coarsest mesh which enabled the critical pressure to converge to within 1% of the exact solution has been chosen. A decomposition into 100 regular elements along the midline ensured the convergence of the elastic buckling load.
The meshes chosen for the sphere and cone cases
Geometry  p COQMULT  q QUA8 

Sphere  100  100 
Cone  50  100 
Linear elastic buckling
Now let us compare the linear elastic buckling loads which were obtained for each case.
A formula for calculating the change in critical pressure for each case when replacing a homogenous shell by a sandwich shell
The function f and the power m depend on the case being considered. Here, one has three cases:

The sphere under external pressure. The critical buckling pressure is given by the formula:${P}_{E}^{\mathit{\text{sphere}}}=\frac{2E}{\sqrt{3\left(1{\nu}^{2}\right)}}{\left(\frac{h}{R}\right)}^{2}$(32)
Substituting z given by Equation 36 into Equation 35, one retrieves Equation 32.

Now let us extend this analytical solution to the case of multilayer spherical shells, which will enable us to predict the analytical critical pressure of the sphere under the assumption that that shell still satisfies shell theory. In order to do that, one replaces the membrane and bending stiffnesses Eh and D in Equation (33) by their homogenized values (E h)^{∗} and D^{∗}. Hence:${p}_{\mathit{\text{crit}}}^{\ast}=\frac{4}{{R}^{2}}\sqrt{{D}^{\ast}{\left(\mathit{\text{Eh}}\right)}^{\ast}}$(37)
This is a very general formula which is valid for any kind of multilayer shell. One can immediately note that the exponent m of interest is equal to $\frac{1}{2}$.
When β = 1 one gets back to r = 1.

The cone under external pressure. The buckling mode of the cone under external pressure fixed along its smaller diameter and free along its larger diameter is similar to that of a cylinder of finite length under external pressure. The proof which leads to the critical loads shows that the buckling pressure is proportional to [(D^{∗})^{3}(E h)^{∗}]^{2.5}. After some calculations, the ratio of the critical loads is found to be:${r}_{\mathit{\text{cone}}\mathit{\text{pext}}}\left(\beta ,x\right)=\frac{{\left[{\left(\beta +x\left[3+3x+{x}^{2}\right]\phantom{\rule{.3em}{0ex}}\right)}^{3}\left(\beta +x\right)\right]}^{0.25}}{{\left(1+x\right)}^{2.5}}$(40)

The cone under internal pressure. In the case of the cone under internal pressure, the critical pressures are estimated by replacing the cone by an equivalent cylinder subjected to uniform axial compression at its base. The proof of the theoretical formulas giving the critical load of the cylinder [12] is more complex than, but similar to that given above for the sphere under external pressure. One can show that the critical load is still governed by $\sqrt{{D}^{\ast}{\left(\mathit{\text{Eh}}\right)}^{\ast}}$. Therefore, this case is similar to that of the sphere under external pressure and the reduction factor is the same:${r}_{\mathit{\text{cone}}\mathit{\text{pint}}}\left(\beta ,x\right)={r}_{\mathit{\text{sphere}}}\left(\beta ,x\right)$(42)
The sphere under external pressure
The calculated buckling loads for the 3 thicknesses
Overall thickness  P_{ E } (Koiter)  COQMULT  QUA8 

mm  (MPa)  (MPa)  (MPa) 
3  2.179  2.179  2.179 
11  29.29  29.12  29.01 
51  630  612  587 
The agreement was very good for the two thinnest shells. The results of the calculation for the thickest shell were still close to the theoretical formula, but less accurate. Let us observe that the thicker the shell, the further the result given by the solid finite element calculation from that given by the COQMULT calculation. The latter also departed from the theoretical solution obtained with Sanders Donnell’s simplified shell theory. Let us note that in the case of the thick shell the slenderness ratio ($\frac{R}{h}$) was only 20: the shell was not really thin and thin shell assumptions were no longer relevant.
Sphere under external pressure: the calculated buckling loads for the different cases
Overall  β  COQMULT  r _{ sphere }  QUA8  Ratio 

thickness  $\frac{{\mathit{P}}_{\mathit{\text{cr}}}}{{\mathit{P}}_{\mathit{\text{euler}}}^{\mathit{E}\mathbf{=}\mathbf{200000}}}$  solution  $\frac{{\mathit{P}}_{\mathit{\text{cr}}}}{{\mathit{P}}_{\mathit{\text{euler}}}^{\mathit{E}\mathbf{=}\mathbf{200000}}}$  QUA8/  
(mm)  COQMULT  
3  0.0001  0.80  0.80  0.022  0.30 
3  0.001  0.80  0.80  0.0  0.56 
3  0.01  0.80  0.80  0.256  0.94 
3  0.1  0.82  0.82  0.82  0.99 
3  1.  1.00  1.0  1.00  1.00 
11  0.0001  0.286  0.286  0.022  0.078 
11  0.001  0.286  0.286  0.073  0.25 
11  0.01  0.294  0.294  0.257  0.87 
11  0.1  0.364  0.364  0.369  0.98 
11  1.  1.00  1.0  1.00  0.996 
51  0.0001  0.065  0.065  0.0015  0.023 
51  0.001  0.066  0.066  0.0092  0.13 
51  0.01  0.075  0.075  0.066  0.85 
51  0.1  0.160  0.160  0.165  0.97 
51  1.  1.00  1.0  1.00  0.96 
The calculated critical loads are given in relation to the elastic buckling load of the shell made of a single material with Young’s modulus 200, 000 MPa.
The analytical formula giving the critical pressure of the sandwich shell was perfectly satisfied by the multilayer shell calculation. However, the "QUA8" calculation, which is not based on the same shell assumptions, led to smaller critical pressures when the modulus of the core layer was less than, or equal to, one hundredth of the modulus of the skin. This overestimation became higher as the core layer became thicker and its stiffness became smaller.
With the COQMULT model, once the modulus became smaller than one hundredth that of the skins, the core layer ceased to contribute to the buckling strength. Nevertheless, the critical load thus estimated still could be much greater than that obtained using a QUA8 model, which allows the two skins to have independent kinematics.
In the case of the thicker shell with the smaller modulus, the critical load was overestimated by a factor of 50 when using the multilayer shell model, which is considerable. The reason was that in the case of the COQMULT model the predicted mode was such that the two skins were interdependent (Figure 4A), whereas with the QUA8 model the predicted buckling mode involved both skins when β = 1, but only the external skin when β < 0.01. This is clearly visible if one compares Figures (4C) and (4D).
The elastic buckling loads for E_{ core } = 0 . 2 MPa
Overall thickness  ${\mathit{P}}_{\mathit{\text{lin}}}^{\mathit{\text{buck}}}$ (skin)  COQMULT  QUA8 

mm  (MPa)  (MPa)  (MPa) 
3  0.242  1.74  0.316 
11  0.242  8.34  0.270 
51  0.242  39.7  0.246 
The multilayer model overestimated the critical bucking load of the assembly by a factor of 100 in the case of the thickest shell, which is considerable.
The elastic buckling loads for E_{ core } = 20 MPa
Overall thickness  ${\mathit{P}}_{\mathit{\text{lin}}}^{\mathit{\text{buck}}}$ (skin)  COQMULT  QUA8 

mm  (MPa)  (MPa)  (MPa) 
3  0.242  1.74  0.55 
11  0.242  8.34  0.65 
51  0.242  39.7  0.9 
The cones under external pressure

Let us first consider the cones meshed with QUA8 elements. In order to do that, the critical loads obtained with two different representations of the same midline will be compared. By meshing the cones with a horizontal free boundary ("HORI") instead of a free surface perpendicular to the the mean surface ("ORTHO") (Figure 6), the buckling pressure could be reduced significantly.
The calculated buckling pressures and modes for the 3 cone angles and the 3 models: Shell, QUA8 with the standard free surface ("ORTHO") and QUA8 with the horizontal free surface ("HORI")
Angle  COQMULT  Mode  Standard QUA8  Mode  Horizontal QUA8  Mode 

°  (MPa)  (MPa)  (MPa)  
30  7.34  4  7.28  4  1.15  5 
45  9.21  4  9.06  4  3.607  5 
60  7.83  4  8.08  4  5.529  4 
From here on, the QUA8 mesh which conforms to the geometry of the COQMULT mesh will be retained.

Now let us compare the results of the COQMULT models with the results of the QUA8 models. Table 7 shows the results of the calculations in the case of singlematerial cones. Again, in this case, one can see that the critical load obtained with the solid model was very close to that given by the shell model for the two thinnest walls, and was slightly smaller for the thickest wall. Once again, the problem lies near the limit of validity of the shell model.
The calculated linear external buckling pressure for the 3 cone angles and the 3 thicknesses
Angle  Overall thickness  COQMULT  Mode  QUA8  Mode 

°  mm  (MPa)  (MPa)  
30  3  0.0052  9  0.0052  9 
30  11  0.139  6  0.139  6 
30  51  7.34  4  7.28  4 
45  3  0.00709  9  0.00708  8 
45  11  0.187  6  0.187  6 
45  51  9.21  4  9.06  5 
60  3  0.00677  8  0.00678  8 
60  11  0.107  6  0.108  6 
60  51  7.83  4  8.08  4 
The typical buckling modes of the 60° and 30° cones are shown in Figure (5).
The calculated normalized external buckling pressure for the 60° cone
Overall  β  COQMULT  r _{conepext}  Mode  QUA8  Mode  Ratio 

thickness  $\frac{{\mathit{P}}_{\mathit{\text{cr}}}}{{\mathit{P}}_{\mathit{\text{COQMULT}}}^{\mathit{E}\mathbf{=}\mathbf{200000}}}$  $\frac{{\mathit{P}}_{\mathit{\text{cr}}}}{{\mathit{P}}_{\mathit{\text{QUA}}8}^{\mathit{E}\mathbf{=}\mathbf{200000}}}$  QUA8/  
(mm)  COQMULT  
3  0.0001  0.889  0.878  8  0.729  8  0.82 
3  0.001  0.889  0.878  8  0.868  8  0.98 
3  0.01  0.891  0.879  8  0.888  8  1.0 
3  0.1  0.899  0.892  8  0.900  8  1.0 
3  1.0  1.  1.  8  1.0  8  1.0 
11  0.0001  0.366  0.360  5  0.176  7  0.49 
11  0.001  0.366  0.361  5  0.325  5  0.91 
11  0.01  0.376  0.367  5  0.358  5  0.98 
11  0.1  0.443  0.411  5  0.429  5  0.99 
11  1.0  1.  1.  6  1.  5  1.02 
51  0.0001  0.097  0.087  3  0.022  6  0.24 
51  0.001  0.098  0.088  3  0.072  4  0.76 
51  0.01  0.11  0.097  3  0.101  4  0.95 
51  0.1  0.196  0.183  4  0.187  4  0.99 
51  1.0  1.  1.  4  1.  4  1.03 
The calculated normalized external buckling pressure for the 45° cone
Overall  β  COQMULT  r _{conepext}  Mode  QUA8  Mode  Ratio 

thickness  $\frac{{\mathit{P}}_{\mathit{\text{cr}}}}{{\mathit{P}}_{\mathit{\text{COQMULT}}}^{\mathit{E}\mathbf{=}\mathbf{200000}}}$  $\frac{{\mathit{P}}_{\mathit{\text{cr}}}}{{\mathit{P}}_{\mathit{\text{QUA}}\mathbf{8}}^{\mathit{E}\mathbf{=}\mathbf{200000}}}$  QUA8/  
(mm)  COQMULT  
3  0.0001  0.889  0.878  9  0.698  9  0.98 
3  0.001  0.889  0.878  9  0.863  9  0.97 
3  0.01  0.891  0.879  9  0.890  9  1.0 
3  0.1  0.900  0.892  9  0.900  9  1.0 
3  1.0  1.  1.  9  1.0  8  1.0 
11  0.0001  0.360  0.360  6  0.152  8  0.42 
11  0.001  0.361  0.361  6  0.316  6  0.87 
11  0.01  0.367  0.367  6  0.361  6  0.98 
11  0.1  0.412  0.430  6  0.425  6  1.03 
11  1.0  1.  1.  6  1.  6  1.0 
51  0.0001  0.089  0.087  4  0.017  4  0.19 
51  0.001  0.090  0.088  4  0.066  4  0.72 
51  0.01  0.099  0.097  4  0.095  4  0.94 
51  0.1  0.143  0.183  4  0.151  4  1.03 
51  1.0  1.  1.  4  1.  5  1.0 
The calculated normalized external buckling pressure for the 30° cone
Overall  β  COQMULT  r _{conepext}  Mode  QUA8  Mode  Ratio 

thickness  $\frac{{\mathit{P}}_{\mathit{\text{cr}}}}{{\mathit{P}}_{\mathit{\text{COQMULT}}}^{\mathit{E}\mathbf{=}\mathbf{200000}}}$  $\frac{{\mathit{P}}_{\mathit{\text{cr}}}}{{\mathit{P}}_{\mathit{\text{QUA}}\mathbf{8}}^{\mathit{E}\mathbf{=}\mathbf{200000}}}$  QUA8/  
(mm)  COQMULT  
3  0.0001  0.890  0.878  9  0.69  9  0.78 
3  0.001  0.890  0.878  9  0.86  9  0.97 
3  0.01  0.890  0.879  9  0.89  9  0.99 
3  0.1  0.900  0.892  9  0.900  9  1.0 
3  1.0  1.  1.  9  1.0  9  1.0 
11  0.0001  0.360  0.360  6  0.150  8  0.41 
11  0.001  0.364  0.361  6  0.314  6  0.86 
11  0.01  0.370  0.367  6  0.362  6  0.97 
11  0.1  0.430  0.43  6  0.423  6  1.0 
11  1.0  1.  1.  6  1.  6  1.0 
51  0.0001  0.096  0.087  4  0.016  9  0.16 
51  0.001  0.096  0.088  4  0.067  4  0.69 
51  0.01  0.104  0.097  4  0.098  4  0.93 
51  0.1  0.185  0.183  4  0.184  4  0.98 
51  1.0  1.  1.  4  1.  4  0.99 
The cones under internal pressure
The calculated relative internal buckling pressure for the 60° cone
Overall  β  COQMULT  r _{conepint}  QUA8  Ratio 

thickness  $\frac{{\mathit{P}}_{\mathit{\text{cr}}}}{{\mathit{P}}_{\mathit{\text{COQMULT}}}^{\mathit{E}\mathbf{=}\mathbf{200000}}}$  $\frac{{\mathit{P}}_{\mathit{\text{cr}}}}{{\mathit{P}}_{\mathit{\text{QUA}}\mathbf{8}}^{\mathit{E}\mathbf{=}\mathbf{200000}}}$  QUA8/  
(mm)  COQMULT  
3  0.0001  0.804  0.801  0.253  0.32 
3  0.001  0.809  0.801  0.452  0.58 
3  0.01  0.811  0.803  0.755  0.94 
3  0.1  0.829  0.823  0.820  1.00 
3  1.0  1.  1.0  1.  1.01 
11  0.0001  0.300  0.287  0.021  0.07 
11  0.001  0.300  0.287  0.072  0.24 
11  0.01  0.307  0.295  0.264  0.86 
11  0.1  0.377  0.365  0.371  0.98 
11  1.0  1.  1.  1.0  1.0 
51  0.0001  0.074  0.067  0.0015  0.02 
51  0.001  0.075  0.068  0.0074  0.10 
51  0.01  0.084  0.077  0.049  0.57 
51  0.1  0.171  0.165  0.167  0.97 
51  1.0  1.  1.  1.0  0.99 
The calculated relative internal buckling pressure for the 45° cone
Overall  β  COQMULT  r _{conepint}  QUA8  Ratio 

thickness  $\frac{{\mathit{P}}_{\mathit{\text{cr}}}}{{\mathit{P}}_{\mathit{\text{COQMULT}}}^{\mathit{E}\mathbf{=}\mathbf{200000}}}$  $\frac{{\mathit{P}}_{\mathit{\text{cr}}}}{{\mathit{P}}_{\mathit{\text{QUA}}\mathbf{8}}^{\mathit{E}\mathbf{=}\mathbf{200000}}}$  QUA8/  
(mm)  COQMULT  
3  0.0001  0.813  0.801  0.25  0.31 
3  0.001  0.813  0.801  0.49  0.60 
3  0.01  0.815  0.803  0.77  0.95 
3  0.1  0.833  0.823  0.83  1.00 
3  1.0  1.  1.  1.0  1.00 
11  0.0001  0.306  0.287  0.021  0.07 
11  0.001  0.306  0.287  0.079  0.26 
11  0.01  0.314  0.295  0.273  0.87 
11  0.1  0.382  0.365  0.378  0.99 
11  1.0  1.  1.  1.0  1.0 
51  0.0001  0.078  0.067  0.0015  0.02 
51  0.001  0.079  0.068  0.0067  0.08 
51  0.01  0.089  0.077  0.0407  0.45 
51  0.1  0.174  0.165  0.17  0.97 
51  1.0  1.  1.  1.0  0.99 
The calculated relative internal buckling pressure for the 30° cone
Overall  β  COQMULT  r _{conepint}  QUA8  Ratio 

thickness  $\frac{{\mathit{P}}_{\mathit{\text{cr}}}}{{\mathit{P}}_{\mathit{\text{COQMULT}}}^{\mathit{E}\mathbf{=}\mathbf{200000}}}$  $\frac{{\mathit{P}}_{\mathit{\text{cr}}}}{{\mathit{P}}_{\mathit{\text{QUA}}\mathbf{8}}^{\mathit{E}\mathbf{=}\mathbf{200000}}}$  QUA8/  
(mm)  COQMULT  
3  0.0001  0.82  0.801  0.26  0.31 
3  0.001  0.82  0.801  0.54  0.67 
3  0.01  0.82  0.803  0.79  0.96 
3  0.1  0.83  0.823  0.83  1.00 
3  1.0  1.  1.0  1.  1.00 
11  0.0001  0.313  0.287  0.022  0.07 
11  0.001  0.314  0.287  0.097  0.31 
11  0.01  0.321  0.295  0.287  0.89 
11  0.1  0.389  0.365  0.385  0.99 
11  1.0  1.  1.0  1.  1.00 
51  0.0001  0.068  0.067  0.0017  0.026 
51  0.001  0.069  0.068  0.0072  0.10 
51  0.01  0.078  0.077  0.044  0.57 
51  0.1  0.160  0.0165  0.162  1.00 
51  1.0  1.  1.0  1.  1.01 
Geometrically non linear elasticplastic buckling
The objective of this section is to answer the question: "Are the limitations of the multilayer shell modeling of sandwich shells observed in the linear case still valid if one takes into account material and geometric nonlinearities?" In order to do that, elasticplastic buckling was predicted by applying the method described previously to the calculation of the stability of the nonlinear elasticplastic solution. A perfect elasticplastic material which becomes plastic when the strain exceeds 0.001 is assumed.
The sphere under external pressure

First, the nonlinear stability analysis was carried out for all the cases.
The calculated elasticplastic buckling loads for the sphere under external pressure
Overall  β  COQMULT  QUA8  Plasticity ratio  

thickness  ${\mathit{P}}_{\mathit{\text{crit}}}^{\mathit{\text{plast}}}$  $\frac{{\mathit{P}}_{\mathit{\text{crit}}}^{\mathit{\text{plast}}}}{{\mathit{P}}_{\mathit{E}}}$  ${\mathit{P}}_{\mathit{\text{crit}}}^{\mathit{\text{plast}}}$  $\frac{{\mathit{P}}_{\mathit{\text{crit}}}^{\mathit{\text{etan}}}}{{\mathit{P}}_{\mathit{E}}}$  QUA8/  
(mm)  (MPa)  (MPa)  COQMULT  
3  0.0001  0.80  0.46  0.08  0.14  0.1 
3  0.01  0.8  0.46  0.77  0.47  0.96 
3  1.  1.17  0.54  1.16  0.53  0.99 
11  0.0001  0.80  0.096  0.095  0.148  0.12 
11  0.01  0.836  0.098  0.833  0.11  1.0 
11  1.  4.4  0.15  4.39  0.15  1. 
51  0.0001  0.802  0.02  0.018  0.021  0.023 
51  0.01  1.0  0.022  0.98  0.025  0.98 
51  1.  20.4  0.033  19.8  0.034  0.97 

Now let us compare the critical loads and complete incremental responses of the COQMULT and QUA8 models in the case of more significant differences. This concerns the spheres with a 49 mmthick core layer (overall thickness 51 mm) and a modulus equal to 20 MPa. This case can be calculated relatively easily because the buckling modes are axisymmetric and, thus, one does not have to carry out coupled Fourier series calculations. Thus, one can use this case to compare the postbuckling behavior given by the two models. The results are summarized in 15. Several effects can be observed. For the COQMULT model with no defect, the two analyses (incremental and bifurcation prediction based upon the tangent modulus theory) led to approximately similar results. In the case of the QUA8 model, the predictions achieved with the three methods – buckling using the tangent modulus (etan), buckling using incremental plasticity theory (flow) and incremental analysis – led to different results. The incremental analysis without defect overestimated the critical load by 50%, except for the case when β = 0.0001. In the other cases, the estimates of the critical loads were identical for the COQMULT and QUA8 models regardless of the bifurcation analysis performed. In the case when β = 0.0001, the multilayer shell analysis overestimated the critical load by 50% when the buckling was not estimated using tangent modulus theory.
Comparison of the estimated elasticplastic buckling loads for the 51 mmthick sphere under external pressure
β  COQMULT  QUA8  

${\mathit{P}}_{\mathit{\text{etan}}}^{\mathit{\text{plast}}}$  ${\mathit{P}}_{\mathit{\text{incr}}}^{\mathit{\text{max}}}$  ${\mathit{P}}_{\mathit{\text{etan}}}^{\mathit{\text{plas}}}$  ${\mathit{P}}_{\mathit{\text{flow}}}^{\mathit{\text{plas}}}$  ${\mathit{P}}_{\mathit{\text{incr}}}^{\mathit{\text{max}}}$  
MPa  MPa  MPA  MPa  MPa  
0.0001  0.802  0.803  0.018  0.555  0.56 
0.01  1.0  1.07  0.98  1.02  1.42 
1.  20.4  20.5  19.8  20.44  33.7 
The cones under external pressure
The calculated elasticplastic buckling load under external pressure for the 60° cone
Overall  β  COQMULT  QUA8  Plasticity ratio  

thickness  ${\mathit{P}}_{\mathit{\text{crit}}}^{\mathit{\text{plast}}}$  $\frac{{\mathit{P}}_{\mathit{\text{crit}}}^{\mathit{\text{plast}}}}{{\mathit{P}}_{\mathit{E}}}$  ${\mathit{P}}_{\mathit{\text{crit}}}^{\mathit{\text{plast}}}$  $\frac{{\mathit{P}}_{\mathit{\text{crit}}}^{\mathit{\text{plast}}}}{{\mathit{P}}_{\mathit{E}}}$  QUA8/  
(mm)  (MPa)  (MPa)  COQMULT  
3  0.0001  0.00605  1.0  0.00496  1.  0.82 
3  0.01  0.00607  1.0  0.00605  1.  1. 
3  1.  0.00859  1.27  0.00681  1.  0.79 
11  0.0001  0.0627  1.0  0.028  0.9  0.44 
11  0.01  0.0642  1.0  0.063  1.  0.98 
11  1.  0.201  1.18  0.158  0.9  0.79 
51  0.0001  0.175  0.23  0.032  0.18  0.18 
51  0.01  0.219  0.25  0.196  0.24  0.89 
51  1.  4.15  0.53  3.55  0.44  0.86 
The calculated elasticplastic buckling load under external pressure for the 45° cone
Overall  β  COQMULT  QUA8  Plasticity ratio  

thickness  ${\mathit{P}}_{\mathit{\text{crit}}}^{\mathit{\text{plast}}}$  $\frac{{\mathit{P}}_{\mathit{\text{crit}}}^{\mathit{\text{plast}}}}{{\mathit{P}}_{\mathit{E}}}$  ${\mathit{P}}_{\mathit{\text{crit}}}^{\mathit{\text{plast}}}$  $\frac{{\mathit{P}}_{\mathit{\text{crit}}}^{\mathit{\text{plast}}}}{{\mathit{P}}_{\mathit{E}}}$  QUA8 /  
(mm)  (MPa)  (MPa)  COQMULT  
3  0.0001  0.0063  1.0  0.00501  1.01  0.79 
3  0.01  0.0063  1.0  0.00634  1.  1.0 
3  1.  0.00898  1.27  0.00715  1.  0.8 
11  0.0001  0.068  1.0  0.0134  0.47  0.20 
11  0.01  0.069  1.0  0.068  1.  0.98 
11  1.  0.212  1.13  0.188  1.  0.89 
51  0.0001  0.144  0.175  0.0159  0.10  0.11 
51  0.01  0.178  0.195  0.154  0.18  0.97 
51  1.  9.21  0.382  2.66  0.294  0.76 
The calculated elasticplastic buckling load under external pressure for the 30° cone
Overall  β  COQMULT  QUA8  Plasticity ratio  

thickness  ${\mathit{P}}_{\mathit{\text{crit}}}^{\mathit{\text{plast}}}$  $\frac{{\mathit{P}}_{\mathit{\text{crit}}}^{\mathit{\text{plast}}}}{{\mathit{P}}_{\mathit{E}}}$  ${\mathit{P}}_{\mathit{\text{crit}}}^{\mathit{\text{plast}}}$  $\frac{{\mathit{P}}_{\mathit{\text{crit}}}^{\mathit{\text{plast}}}}{{\mathit{P}}_{\mathit{E}}}$  QUA8/  
(mm)  (MPa)  (MPa)  COQMULT  
3  0.0001  0.0046  1.0  0.0032  0.9  0.70 
3  0.01  0.0046  1.0  0.0047  1.01  1.0 
3  1.  0.0066  1.27  0.0054  1.04  0.82 
11  0.0001  0.051  1.0  0.00708  0.37  0.15 
11  0.01  0.052  1.0  0.051  1.02  0.98 
11  1.  0.159  1.14  0.14  1.01  0.89 
51  0.0001  0.097  0.14  0.01  0.08  0.10 
51  0.01  0.12  0.16  0.104  0.15  0.87 
51  0.1  2.58  0.35  1.75  0.24  0.68 
For the cones under external pressure, one can note that the buckling was elastic for the two thinnest configurations. For the thickest shell, the buckling was elasticplastic. No significant nonlinear geometric effect was observed for this type of buckling. Once again, when the modulus of the core layer was very small, the buckling mode was a skin mode. This effect cannot be predicted using the COQMULT model. Because of elasticplasticity, the thicker the cone, the smaller the ratio of elasticplastic to elastic buckling pressure. These results confirm those obtained previously in the case of the sphere under external pressure. For the 30° case and a very low stiffness of the core layer, the critical load predicted by the elasticplastic solid model was 10 times smaller.
The cones under internal pressure
The calculated elasticplastic buckling load under internal pressure for the 60° cone
Overall  β  COQMULT  QUA8  Plasticity ratio  

thickness  ${\mathit{P}}_{\mathit{\text{crit}}}^{\mathit{\text{plast}}}$  $\frac{{\mathit{P}}_{\mathit{\text{crit}}}^{\mathit{\text{plast}}}}{{\mathit{P}}_{\mathit{E}}}$  ${\mathit{P}}_{\mathit{\text{crit}}}^{\mathit{\text{plast}}}$  $\frac{{\mathit{P}}_{\mathit{\text{crit}}}^{\mathit{\text{plast}}}}{{\mathit{P}}_{\mathit{E}}}$  QUA8 /  
(mm)  (MPa)  (MPa)  COQMULT  
3  0.0001  0.156  0.3  0.0186  0.11  0.12 
3  0.01  0.158  0.3  0.15  0.31  0.95 
3  1.  0.307  0.48  0.224  0.35  0.73 
11  0.0001  0.175  0.062  0.0306  0.15  0.18 
11  0.01  0.183  0.064  0.142  0.06  0.78 
11  1.  1.04  0.11  0.892  0.09  0.86 
51  0.0001  0.183  0.01  0.0039  0.01  0.02 
51  0.01  0.253  0.012  0.183  0.016  0.72 
51  1.  4.64  0.019  3.3  0.014  0.71 
The calculated elasticplastic buckling load under internal pressure for the 45° cone
Overall  β  COQMULT  QUA8  Plasticity ratio  

thickness  ${\mathit{P}}_{\mathit{\text{crit}}}^{\mathit{\text{plast}}}$  $\frac{{\mathit{P}}_{\mathit{\text{crit}}}^{\mathit{\text{plast}}}}{{\mathit{P}}_{\mathit{E}}}$  ${\mathit{P}}_{\mathit{\text{crit}}}^{\mathit{\text{plast}}}$  $\frac{{\mathit{P}}_{\mathit{\text{crit}}}^{\mathit{\text{plast}}}}{{\mathit{P}}_{\mathit{E}}}$  QUA8/  
(mm)  (MPa)  (MPa)  COQMULT  
3  0.0001  0.127  0.35  0.04  0.37  0.33 
3  0.01  0.128  0.35  0.11  0.31  0.85 
3  1.  0.248  0.55  0.18  0.39  0.71 
11  0.0001  0.145  0.07  0.01  0.07  0.07 
11  0.01  0.152  0.07  0.12  0.06  0.79 
11  1.  0.860  0.125  0.59  0.09  0.69 
51  0.0001  0.145  0.01  0.006  0.02  0.04 
51  0.01  0.184  0.01  0.048  0.006  0.26 
51  1.  3.80  0.02  0.77  0.004  0.2 
The calculated elasticplastic buckling load under internal pressure for the 30° cone
Overall  β  COQMULT  QUA8  Plasticity ratio  

thickness  ${\mathit{P}}_{\mathit{\text{crit}}}^{\mathit{\text{plast}}}$  $\frac{{\mathit{P}}_{\mathit{\text{crit}}}^{\mathit{\text{plast}}}}{{\mathit{P}}_{\mathit{E}}}$  ${\mathit{P}}_{\mathit{\text{crit}}}^{\mathit{\text{plast}}}$  $\frac{{\mathit{P}}_{\mathit{\text{crit}}}^{\mathit{\text{plast}}}}{{\mathit{P}}_{\mathit{E}}}$  QUA8 /  
(mm)  (MPa)  (MPa)  COQMULT  
3  0.0001  0.086  0.44  0.004  0.06  0.05 
3  0.01  0.087  0.44  0.06  0.34  0.75 
3  1.  0.165  0.69  0.11  0.48  0.70 
11  0.0001  0.10  0.09  0.01  0.09  0.09 
11  0.01  0.11  0.09  0.08  0.75  0.75 
11  1.  0.61  0.16  0.39  0.64  0.64 
51  0.0001  0.11  0.015  0.011  0.06  0.10 
51  0.01  0.13  0.016  0.095  0.02  0.71 
51  0.1  2.70  0.025  1.19  0.01  0.44 
In this case, one finds again, although to a lesser degree than for the sphere, that tangent modulus theory led to a safe estimate of the critical load for β = 0.0001, the QUA8 model and the thickest shell. For the other cases, the COQMULT became more unsafe as the shell’s thickness increased. The critical plastic buckling pressure predicted using incremental plasticity theory along with the QUA8 model was up to 50% greater than using tangent modulus theory. However, both these critical loads were always smaller than, or equal to, that obtained using the COQMULT model.
Conclusions
This paper presented the analytical formulae for the elastic buckling loads of sandwich cones and spheres under pressure loading. These formulae were successfully compared to multilayer shell calculations. Then, one showed the limitations of that formulation which relies on a shell kinematics which does not allow the skins to wrinkle independently of each other. The calculations were compared to continuum mechanics calculations, which are the most relevant calculations in these cases.
All these calculations showed that when the Young’s modulus of the sandwich shell’s core layer is more than one tenth that of the skins the multilayer model suffices. When the ratio of the moduli is less than one hundredth (which is often the case with foam materials) skin modes are likely to occur at pressures which are much lower than those predicted by the multilayer model. These conclusions are valid both in elasticity and in nonlinear elasticplasticity. The analysis which is proposed here for the axisymmetric case is particularly efficient since a linear calculation such as those which were carried out in this study takes no more than a few seconds on a laptop PC and a nonlinear solid calculation takes only a few minutes.
Declarations
Authors’ Affiliations
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